Fidelity estimation for quantum computing systems

ABSTRACT

Methods and apparatus for estimating the fidelity of quantum hardware. In one aspect, a method includes accessing a set of quantum gates; sampling a subset of quantum gates from the set of quantum gates, wherein the subset of quantum gates defines a quantum circuit; applying the quantum circuit to a quantum system and performing measurements on the quantum system to determine output information of the quantum system; calculating output information of the quantum system based on application of the quantum circuit to the quantum system; and estimating a fidelity of the quantum circuit based on the determined output information and the calculated output information of the quantum system.

BACKGROUND

This specification relates to quantum computing.

Quantum circuits are models for quantum computation in which acomputation includes a sequence of quantum gates. Quantum circuits aresensitive to errors, e.g., due to decoherence and other quantum noise.The effect of errors in a quantum circuit may be characterized by thefidelity of the quantum circuit. Fidelity is a metric of quantumcircuits that indicates the quality and reliability of a quantumcircuit.

SUMMARY

This specification relates to estimating the fidelity of quantumhardware in quantum computing systems. In particular, this specificationdescribes technologies for estimating the fidelity of complex,non-Clifford quantum circuits with multiple qubits.

In general, one innovative aspect of the subject matter described inthis specification can be implemented in a method that includes theactions of accessing a set of quantum gates, sampling a subset ofquantum gates from the set of quantum gates, wherein the subset ofquantum gates defines a quantum circuit, applying the quantum circuit toa quantum system and performing measurements on the quantum system todetermine output information of the quantum system, calculating outputinformation of the quantum system based on application of the quantumcircuit to the quantum system, and estimating a fidelity of the quantumcircuit based on the determined output information and the calculatedoutput information of the quantum system.

Other implementations of this aspect include corresponding computersystems, apparatus, and computer programs recorded on one or morecomputer storage devices, each configured to perform the actions of themethods. A system of one or more computers can be configured to performparticular operations or actions by virtue of having software, firmware,hardware, or a combination thereof installed on the system that inoperation causes or cause the system to perform the actions. One or morecomputer programs can be configured to perform particular operations oractions by virtue of including instructions that, when executed by dataprocessing apparatus, cause the apparatus to perform the actions.

The foregoing and other implementations can each optionally include oneor more of the following features, alone or in combination. In someimplementations estimating a fidelity of the quantum system comprisesfitting the determined output information of the quantum system to thecalculated output information of the quantum system.

In some cases fitting the determined output information of the quantumsystem to the calculated output information of the quantum system toestimate the fidelity of the quantum circuit comprises: defining aconvex combination of the calculated output information of the quantumsystem and a totally mixed quantum state; and estimating the fidelity ofthe quantum circuit by comparing the defined convex combination with thedetermined output information of the quantum system.

In some implementations the convex combination is given by

ρ = α ⁢  ψ 〉 ⁢ 〈 ψ  + ( 1 - α ) ⁢ Nwherein α represents the fidelity of the quantum circuit, |ψ

represents a calculated quantum state of the quantum system based onapplication of the quantum circuit to the quantum system, and

/N represents the totally mixed state.

In some cases the method further comprises repeatedly sampling a subsetof quantum gates from the set of quantum gates until completion of anevent, wherein each subset of quantum gates defines a respective quantumcircuit; for each sampled subset of quantum gates: applying therespective quantum circuit to a quantum system and performing respectivemeasurements on the quantum system to determine output information ofthe quantum system; calculating output information of the quantum systembased on application of the respective quantum circuit to the quantumsystem; and estimating a fidelity of the respective quantum circuitbased on the determined output information and the calculated outputinformation of the quantum system.

In some implementations the completion of the event occurs when anuncertainty of an estimated fidelity is below a predetermined threshold.

In some cases the set of quantum gates comprises a universal set ofquantum gates.

In some cases the set of quantum gates comprise single qubit quantumgates and two qubit quantum gates.

In some implementations each gate in the set of quantum gates isassociated with a respective quantum gate fidelity.

In other implementations the sampled subset of quantum gates comprises asame number of quantum gates of comparable respective quantum gatefidelity.

In some cases sampling a subset of quantum gates from the set of quantumgates comprises randomly sampling a subset of quantum gates from the setof quantum gates.

The subject matter described in this specification can be implemented inparticular ways so as to realize one or more of the followingadvantages.

Quantum hardware, e.g., a system of quantum gates, is inherently proneto errors that need to be characterized before they can be corrected.Full characterization via processes such as quantum process tomographyare impractical, e.g., in terms of computational cost and efficiency.For example, quantum process tomography becomes prohibitive as thenumber of qubits in the quantum system grows because the number ofrequired measurements grows exponentially. In addition, fullcharacterization is often unnecessary since for practical purposes, itmay be enough to estimate more general quantities such as the averagefidelity.

Alternative methods for characterizing errors include using arestrictive set of quantum gates in the quantum hardware. For example,randomized benchmarking with Clifford gates is a widely extended methodto measure the fidelity of single qubit gates and two-qubit gates.However, such techniques cannot be applied to directly measure thefidelity of quantum circuits which employ a universal quantum gate set.The results obtained using such methods may therefore not be ofimmediate interest or considered illustrative and impractical since thequantum states produced by these restrictive families of circuits may bevery different in important aspects to quantum states produced bycircuits with universal quantum gates. For example, Clifford circuitsmay always be efficiently simulated using classical computers, and donot exhibit the Porter-Thomas distribution.

As the complexity and variety of quantum hardware grows, it isimperative to measure the circuit fidelity for quantum circuits that arenot of a type which can be simulated easily and efficiently by classicalcomputers.

A system implementing fidelity estimation for quantum computing systemsas described in this specification is able to estimate the fidelity ofincreasingly complex quantum hardware.

A system implementing fidelity estimation for quantum computing systemsas described in this specification is applicable to both digital modelsof quantum computation and analog models of quantum computation. In thecase of digital models of quantum computation, the system does notrequire the use of a restrictive set of quantum gates. For example, thesystem may utilize complex random quantum circuits composed of gateschosen from a universal set of quantum gates, thus allowing for directmeasurement of the quantum circuit fidelity for increasingly complexquantum circuits with an increasing number of qubits and quantum gates.Similarly, in the case of analog models of quantum computation, thesystem allows for directly determining the fidelity of quantum hardwareimplementing continuous Hamiltonian evolution.

Unlike other systems and methods for estimating the fidelity of aquantum circuit, a system implementing fidelity estimation for quantumcomputing systems as described in this specification enables thefidelity of a quantum circuit to be estimated without requiring that anumber of measurements exponential in the number of qubits be performed.

The details of one or more implementations of the subject matter of thisspecification are set forth in the accompanying drawings and thedescription below. Other features, aspects, and advantages of thesubject matter will become apparent from the description, the drawings,and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A and 1B depict example systems for fidelity estimation.

FIG. 2 is a flow diagram of an example process for benchmarking thefidelity of a quantum circuit.

FIG. 3 is a flow diagram of an example process for fitting determinedoutput information of a quantum system to calculated output informationof the quantum system to estimate a fidelity of a quantum circuit.

FIG. 4 is a flow diagram of an example process for benchmarking thefidelity of quantum hardware.

FIG. 5 is a flow diagram of an example process for fitting determinedoutput information of a quantum system to calculated output informationof the quantum system to estimate a fidelity of a quantum hardware.

Like reference numbers and designations in the various drawings indicatelike elements.

DETAILED DESCRIPTION

This specification describes methods and systems for fidelitybenchmarking for quantum hardware, e.g., quantum circuits. A randominstance of a quantum circuit from a given universal family of availablequantum gates is selected, and the statistics of the selected quantumcircuit for a chosen measurement are numerically calculated usingclassical simulation. A sequence of runs of the same quantum circuit isperformed and measurements on the quantum hardware are performed. Thefidelity of the quantum circuit is estimated using the numericallyobtained expected statistics and experimentally determined statistics.

Example Operating Environment

FIG. 1A depicts an example system 100 for fidelity estimation. Theexample system 100 is an example of a system implemented as classical orquantum computer programs on one or more classical computers or quantumcomputing devices in one or more locations, in which the systems,components, and techniques described below can be implemented.

The system includes quantum hardware 102 in communication with afidelity estimation system 104. The quantum hardware 102 includes aquantum system that may include one or more qubits 106, e.g., qubit 108.The one or more qubits may be used to perform algorithmic operations orquantum computations. The specific realization of the one or more qubitsdepends on the type of algorithmic operations or quantum computationsthat the quantum computing device is performing. For example, the qubitsmay include qubits that are realized via atomic, molecular orsolid-state quantum systems. In other examples the qubits may include,but are not limited to, superconducting qubits or semi conductingqubits. For clarity, four qubits are depicted in FIG. 1A, however thesystem may include a smaller or larger number of qubits.

Each of the one or more qubits 106 may interact with one or more otherqubits, e.g., through respective controllable couplings. In someexamples the one or more qubits 106 may be subject to nearest neighborinteractions.

The one or more qubits 106 may be arranged in a variety of ways. Thespecific arrangement of the one or more qubits may depend on thealgorithmic operation or quantum computation that the qubits are beingused to perform. In some examples the qubits may be arranged in aone-dimensional array, e.g., a chain. In other examples the qubits maybe arranged in a two-dimensional array, e.g., a lattice. For clarity,four qubits are depicted in a one-dimensional array in FIG. 1, howeverthe system may arrange the qubits in other fashions.

The quantum hardware 102 includes a set of quantum gates 110. The set ofquantum gates 110 includes single qubit gates, e.g., quantum gate 112,and two-qubit gates, e.g., quantum gate 114. Single qubit quantum gatesare quantum gates that operate on a single qubit. Example single qubitgates include but are not limited to Hadamard gates, Pauli X, Y or Zgates, or phase shift gates. Two-qubit quantum gates are quantum gatesthat operate on two qubits. Example two-qubit gates include but are notlimited to swap gates, controlled gates, Toffoli gates or Fredkin gates.

The set of quantum gates 110 may include a universal set of quantumgates. A universal set of quantum gates is a set of gates to which anycomputational operation possible on a quantum computing device can bereduced. For example, one example of a universal set of single andtwo-qubit quantum gates includes a Hadamard gate, π/8 gate andcontrolled NOT gate.

The set of quantum gates 110 may be sampled to define one or morequantum circuits, e.g., quantum circuit 116. For clarity, quantumcircuit 116 includes a fixed number of representative quantum gates,e.g., quantum gates 112 and 114, however a quantum circuit defined by asampled set of quantum gates may include more or fewer quantum gates invarying arrangements.

The sampled quantum circuit 116 receives as input a quantum system,e.g., one or more qubits, prepared in an initial state 122, e.g., aground state. The quantum circuit operates on the quantum system andoutputs the quantum system in a final state 120, wherein the final stateis determined by the operations performed on the quantum system by thequantum circuit.

Each quantum gate in the set of quantum gates 110 and each quantumcircuit defined by subsets of the set of quantum gates 110 is associatedwith a respective gate fidelity or quantum circuit fidelity. A quantumgate fidelity and quantum circuit fidelity indicates a measure of howreliably the gate or circuit transforms an input into an expectedoutput. For example, a Pauli X quantum gate acts on a single qubit andmaps a zero state to a one state, and a one state to a zero state. Afidelity of a Pauli X gate may include a number between 0 and 1 thatindicates how accurately and reliably the mapping is achieved, e.g.,whether the gate reliably maps a zero state to a one state and viceversa.

The quantum hardware 102 includes one or more measurement devices 118,e.g., measurement device 124. The measurement devices 118 may operate onthe quantum system to determine properties of the quantum system, e.g.,measurement device 124 operates on the quantum system in the final state120.

The fidelity estimation system 104 may include a classical or quantumprocessing device and communicates with the quantum hardware 102. Thefidelity estimation system 104 may be configured to access the set ofquantum gates 110 and sample a subset of quantum gates from the set ofquantum gates 110 to define a respective quantum circuit, e.g., quantumcircuit 116. The fidelity estimation system 104 may cause a definedquantum circuit to be repeatedly applied to a quantum system, e.g., oneor more qubits 106, and perform respective measurements on the quantumsystem to determine output information, e.g., statistics, of a quantumsystem using one or more measurement devices 118.

The fidelity estimation system 104 may be configured to calculate outputinformation, e.g., expected statistics, of the quantum system based onapplication of the quantum circuit to the quantum system, e.g., afterapplication of a defined quantum circuit on the quantum system, and tofit determined output information of the quantum system to thecalculated output information of the quantum system to estimate afidelity of the quantum circuit. Estimating a fidelity of a quantumcircuit is described in more detail below with reference to FIGS. 2 and3.

FIG. 1B depicts an example system 150 for fidelity estimation. Theexample system 150 is an example of a system implemented as classical orquantum computer programs on one or more classical computers or quantumcomputing devices in one or more locations, in which the systems,components, and techniques described below can be implemented.

The system includes quantum hardware 152 in communication with afidelity estimation system 154. The quantum hardware 152 includes aquantum system that may include one or more qubits 156, e.g., qubit 158.As described above with reference to FIG. 1A, the one or more qubits maybe used to perform algorithmic operations or quantum computations. Thespecific realization of the one or more qubits depends on the type ofalgorithmic operations or quantum computations that the quantumcomputing device is performing. For example, the qubits may includequbits that are realized via atomic, molecular or solid-state quantumsystems. In other examples the qubits may include, but are not limitedto, superconducting qubits or semi conducting qubits. For clarity, fourqubits are depicted in FIG. 1B, however the system may include a smalleror larger number of qubits.

Each of the one or more qubits 156 may interact with one or more otherqubits, e.g., through respective controllable couplings. In someexamples the one or more qubits 156 may be subject to nearest neighborinteractions.

The one or more qubits 156 may be arranged in a variety of ways. Thespecific arrangement of the one or more qubits may depend on thealgorithmic operation or quantum computation that the qubits are beingused to perform. In some examples the qubits may be arranged in aone-dimensional array, e.g., a chain. In other examples the qubits maybe arranged in a two-dimensional array, e.g., a lattice. For clarity,four qubits are depicted in a one-dimensional array in FIG. 1, howeverthe system may arrange the qubits in other fashions.

The quantum hardware 152 includes one or more components for performingcontinuous Hamiltonian evolution 160. The one or more components forperforming continuous Hamiltonian evolution 160 implement one or moreHamiltonians, which in turn may determine a unitary operator thatdefines the evolution of the quantum system on which it is applied. Forexample, the components 160 may implement a Hamiltonian H, which in turngives rise to the unitary operator 162. The unitary operator 162 definesan evolution of the quantum system prepared in initial state 164,resulting in a final state of the quantum system 166.

Each continuous Hamiltonian evolution of the quantum system isassociated with a respective quantum hardware fidelity. Quantum hardwarefidelity indicates a measure of how reliably the hardware transforms aninput into an expected output, as described above with reference to FIG.1A.

The quantum hardware 152 includes one or more measurement devices 168,e.g., measurement device 172. The measurement devices 168 may operate onthe quantum system to determine properties of the quantum system, e.g.,measurement device 172 operates on the quantum system in the final state166.

The fidelity estimation system 154 may include a classical or quantumprocessing device and communicates with the quantum hardware 152. Thefidelity estimation system 154 may be configured to access the quantumhardware 152 and select components for particular continuous Hamiltonianevolutions 160. The fidelity estimation system 154 may cause the quantumhardware to repeatedly perform a continuous Hamiltonian evolutioncorresponding to selected components on the quantum system, e.g., theone or more qubits 156, and perform respective measurements on thequantum system to determine output information of the quantum system.

The fidelity estimation system 154 may be configured to calculate outputinformation of the quantum system based on performing the selectedcontinuous Hamiltonian evolution on the quantum system and fit thedetermined output information of the quantum system to the calculatedoutput information of the quantum system to estimate the fidelity of thequantum hardware. Estimating a fidelity of quantum hardware is describedin more detail below with reference to FIGS. 4 and 5.

Programming the Hardware

FIG. 2 is a flowchart of an example process 200 for benchmarking thefidelity of a quantum circuit. For convenience, the process 200 will bedescribed as being performed by a system of one or more classical orquantum computing devices located in one or more locations. For example,a quantum computing system, e.g., the quantum computing system 100 ofFIG. 1A, appropriately programmed in accordance with this specification,can perform the process 200.

The system accesses a set of quantum gates (step 202). The set ofquantum gates may include one or more single qubit gates. Each quantumgate in the set of quantum gates may be associated with a respectivequantum gate fidelity. The set of quantum gates may include one or moretwo-qubit gates. Example single and two-qubit gates are described abovewith reference to FIG. 1A. In some implementations the set of quantumgates includes a universal set of quantum gates. Universal sets ofquantum gates are described above with reference to FIG. 1A.

The system samples a subset of quantum gates from the set of quantumgates (step 204). In some implementations the system may randomly samplea subset of quantum gates from the set of quantum gates. The sampledsubset of quantum gates defines a quantum circuit. For example, byrandomly sampling a subset of quantum gates from the available set ofquantum gates the system produces a random instance of a quantumcircuit.

In some implementations the system may repeatedly sample subsets ofquantum gates from the set of quantum gates, where each sampled subsetdefines a respective quantum circuit. The system may sample subsets ofquantum gates that include a same number of quantum gates of comparablerespective quantum gate fidelity, e.g., the instance of the quantumcircuit defined by the sampled subset of quantum gates may include asame number of quantum gates of comparable quantum gate fidelity. Bysampling subsets of quantum gates that include a same number of quantumgates of comparable respective quantum gate fidelity, the system is ableto improve the consistency of the results obtained by the process 200and avoid large systematic errors.

The system applies the quantum circuit to a quantum system and performsmeasurements on the quantum system to determine output information,e.g., statistics of the quantum system (step 206). For example, thesystem may include or otherwise access a quantum system, e.g., a quantumsystem including one or more qubits as illustrated in FIG. 1A, andrepeatedly apply the quantum circuit defined by the sampled subset ofquantum gates to an initialized state of the quantum system. For eachapplication of the quantum circuit, the system may perform respectivemeasurements on the quantum system and use the measurement results todetermine output information of the quantum system. As an example, thesystem may sample a subset of quantum gates that define a respectivequantum circuit, and perform m runs of the quantum circuit on a quantumsystem of interest to obtain a sequence of bit-strings {x₁, x₂, . . . ,x_(m)} measured in the computational basis.

As described above, in some implementations the system may repeatedlysample subsets of quantum gates from the set of quantum gates, whereeach sampled subset defines a respective quantum circuit. In such casesthe system may repeatedly applies each sampled quantum circuit to thequantum system and performs respective measurements on the quantumsystem to determine respective statistics of the quantum system for eachof the sampled circuits.

The system calculates output information, e.g., expected, e.g., exact orideal, statistics, of the quantum system based on application of thequantum circuit to the quantum system (step 208). For example, thesystem may use available computing technology, e.g., classical computingtechnology, to calculate the output information of the quantum systemafter application of the quantum circuit to the quantum system.

Continuing the example above, in order to determine output informationof the quantum system the system may calculate a set of probabilities{p(y_(j))} that correspond to the probability of obtaining each possiblebit string y_(j). As described above with reference to step 204, in someimplementations the system may repeatedly sample subsets of quantumgates from the set of quantum gates, where each sampled subset defines arespective quantum circuit. In such cases the system may calculate a setof probabilities that correspond to the probability of obtaining eachpossible bit string for each sampled quantum circuit.

The system estimates a fidelity of the quantum circuit based on thedetermined output information and the calculated output information ofthe quantum system (step 210). The system may estimate the fidelity ofthe quantum circuit by fitting the determined output information of thequantum system to the calculated output information of the quantumsystem. The system may fit the results of performing the measurements instep 206 to a statistical mixture of (i) the output information of thequantum system based on application of the quantum circuit to thequantum system as calculated in step 208, and (ii) a totally mixedquantum state. Fitting determined output information of a quantum systemto calculated output information of the quantum system to estimate afidelity of a quantum circuit is described in more detail below withreference to FIG. 3.

In cases where the system repeatedly samples subsets of quantum gates todefine multiple quantum circuits, the system fits the respectivedetermined output information for the quantum system to the respectivecalculated output information of the quantum system to estimate arespective fidelity of each quantum circuit. The system may repeatedlysample subsets of quantum gates from the set of quantum gates untilcompletion of an event, e.g., when an uncertainty of an estimatedfidelity is below a predetermined threshold. By increasing the number ofrepetitions, the uncertainty of the estimated fidelity may be reduced,e.g., following an inverse square root law in the number of repetitions,to a desired threshold of certainty.

FIG. 3 is a flowchart of an example process 300 for fitting determinedoutput information, e.g., statistics, of a quantum system to calculatedoutput information, e.g., statistics, of the quantum system to estimatea fidelity of a quantum circuit. For convenience, the process 300 willbe described as being performed by a system of one or more classical orquantum computing devices located in one or more locations. For example,one or more classical processors, e.g., classical processors 104 of FIG.1A, appropriately programmed in accordance with this specification, canperform the process 300.

The system defines a convex combination of the calculated outputinformation of the quantum system described above with reference to step208 of FIG. 2 and a totally mixed quantum state (step 302). For example,the convex combination may be given by equation (1) below.

ρ = α ⁢  ψ 〉 ⁢ 〈 ψ  + ( 1 - α ) ⁢ N ( 1 )

In equation (1), a represents the fidelity of the quantum circuit, |ψ

represents a calculated quantum state of the quantum system based onapplication of the quantum circuit to the quantum system, and

/N represents the totally mixed state with N a dimension of the Hilbertspace containing |ψ

.

The system estimates the fidelity of the quantum circuit by comparingthe defined convex combination in equation (1) with the determinedoutput information of the quantum system described above with referenceto step 206 of FIG. 2 (step 304). For example, the system may estimatethe fidelity of the quantum circuit by comparing the defined convexcombination with the determined output information of the quantum systemand solving for the parameter α.

Continuing the example provided above in FIG. 2, the system maynumerically calculate the quantity

$c = {\frac{1}{m}{\sum\limits_{j = 1}^{m}\;{\ln\left( {p\left( x_{j} \right)} \right)}}}$for each sampled subset of quantum gates for a correspondingexperimentally obtained sequence of bit-strings {x₁, x₂, . . . x_(m)}.Under certain assumptions, e.g., assuming that the quantum circuit islong enough (the depth of the circuit may grow no faster than n^(1/D),up to a possible correction in log(n), where n is the number of qubitsand D is the dimension of the qubit array, e.g., D=1 and depth=n for a1D array of qubits, as depicted in FIG. 1A, or D may be infinite withconstant depth or depth logarithmic in n for hypothetical configurationswhere two qubit gates are performed between any pairs of qubits), theparameter α, e.g., the quantum circuit fidelity, may be estimated byα=c+ln(N)+γ where γ is the Euler constant and c is defined above. Insome implementations the error in the estimation of α may be given byk/m^(1/2), where k≅1. This may represent a number of requiredmeasurements, and is independent of the number of qubits.

In some implementations the system may estimate a fidelity of a quantumcircuit through numerical comparison with any number of statisticalaggregates. For example, instead of using the quantity Σ_(j=1) ^(m)ln(p(x_(j))) as in the above, the system may use the quantity Σ_(j=1)^(m)p(x_(j))², e.g., the sum of the squares of the probabilities. Otherquantities may also be used. An essential requirement is that thequantity be a statistical aggregate that may be computed using asimulation of the quantum circuit, and that the quantity must be equallysensitive to errors in the quantum circuit physical implementation.

As described above, in some implementations the system may repeatedlysample subsets of quantum gates and repeatedly estimate correspondingfidelities for quantum circuits defined by the subsets of quantum gates.By repeatedly estimating fidelities for the quantum circuits, the systemis able to increase the reliability of the fidelity estimate and reducethe likelihood of systematic errors or correlations affecting obtainedresults. As a simple example, by repeatedly estimating fidelities, thesystem is able to determine that the quantum hardware is functioning asintended.

FIG. 4 is a flowchart of an example process 400 for benchmarking thefidelity of quantum hardware. For convenience, the process 400 will bedescribed as being performed by a system of one or more classical orquantum computing devices located in one or more locations. For example,a quantum computing system, e.g., the quantum computing system 150 ofFIG. 1B, appropriately programmed in accordance with this specification,can perform the process 400.

The system accesses quantum hardware (step 402). The quantum hardwaremay be configured to perform one or more different continuousHamiltonian evolutions, as described above with reference to FIG. 1B.Each continuous Hamiltonian evolution may be associated with arespective quantum hardware fidelity.

The system selects a particular continuous Hamiltonian evolution (step404). For example, the system may select one or more components that areconfigured to implement a particular Hamiltonian that determines aunitary operator that defines an evolution of the quantum system onwhich it is applied. In some cases the system may randomly select aparticular continuous Hamiltonian evolution.

In some implementations the system may repeatedly select continuousHamiltonian evolutions. For example, in some cases the quantum hardwaremay be configured to perform one or more different continuousHamiltonian evolutions on one or more interacting qubits where eachqubit interaction has an associated respective fidelity. In these cases,the system may repeatedly select continuous Hamiltonian evolutions thatinclude interactions of comparable fidelity. By selecting continuousHamiltonian evolutions including qubit interactions of comparablefidelity, the system is able to improve the consistency of the resultsobtained by the process 400 and avoid large systematic errors.

The system performs the selected continuous Hamiltonian evolution of aquantum system and performs measurements on the quantum system todetermine output information of the quantum system (step 406). Forexample, the system may include or otherwise access a quantum system,e.g., a quantum system including one or more qubits as illustrated inFIG. 1B, and repeatedly perform the selected continuous Hamiltonianevolution by selecting one or more corresponding components thatimplement a Hamiltonian which gives rise to a unitary operator thatdefines the evolution of the quantum system, as described above withreference to FIG. 1B. For each evolution of the quantum system, thesystem may perform respective measurements on the quantum system and usethe measurement results to determine output information of the quantumsystem.

As described above, in some implementations the system may repeatedlyselect continuous Hamiltonian evolutions. In such cases the system mayrepeatedly performs each selected continuous Hamiltonian evolution ofthe quantum system and performs respective measurements on the quantumsystem to determine respective output information of the quantum systemfor the selected continuous Hamiltonian evolutions.

The system calculates output information of the quantum system based onperforming the selected continuous Hamiltonian evolution on the quantumsystem (step 408). For example, the system may use available computingtechnology, e.g., classical computing technology, to calculate theoutput information of the quantum system after the selected continuousevolution of the quantum system.

The system estimates a fidelity of the quantum hardware based on thedetermined output information and the calculated output information ofthe quantum system (step 410). The system may estimate the fidelity ofthe quantum hardware by fitting the determined output information of thequantum system to the calculated output information of the quantumsystem. The system may fit the results of performing the measurements instep 406 to a statistical mixture of (i) the output information of thequantum system based on performing the selected continuous Hamiltonianevolution on the quantum system as calculated in step 408, and (ii) atotally mixed quantum state. Fitting determined output information of aquantum system to calculated output information of the quantum system toestimate a fidelity of quantum hardware is described in more detailbelow with reference to FIG. 5.

In cases where the system repeatedly selects continuous Hamiltonianevolutions, the system fits the respective determined output informationfor the quantum system to the respective calculated output informationof the quantum system to estimate a respective fidelity of each quantumhardware corresponding to a respective selected continuous Hamiltonianevolution. The system may repeatedly select continuous Hamiltonianevolutions until completion of an event, e.g., when an uncertainty of anestimated fidelity is below a predetermined threshold. By increasing thenumber of repetitions, the uncertainty of the estimated fidelity may bereduced, e.g., following an inverse square root law in the number ofrepetitions, to a desired threshold of certainty.

FIG. 5 is a flowchart of an example process 500 for fitting determinedoutput information of a quantum system to calculated output informationof the quantum system to estimate a fidelity of a quantum hardware. Forconvenience, the process 500 will be described as being performed by asystem of one or more classical or quantum computing devices located inone or more locations. For example, one or more classical processors,e.g., classical processors 104 of FIG. 1, appropriately programmed inaccordance with this specification, can perform the process 500.

The system defines a convex combination of the calculated outputinformation of the quantum system described above with reference to step408 of FIG. 4 and a totally mixed quantum state (step 502). For example,the convex combination may be given by equation (1) below.

ρ = α ⁢  ψ 〉 ⁢ 〈 ψ  + ( 1 - α ) ⁢ N ( 1 )

In equation (1), a represents the fidelity of the quantum circuit, |ψ

represents a calculated quantum state of the quantum system based onperforming the continuous Hamiltonian evolution on the quantum system,and

/N represents the totally mixed state with N a dimension of the Hilbertspace containing |ψ

.

The system estimates the fidelity of the quantum hardware by comparingthe defined convex combination in equation (1) with the determinedoutput information of the quantum hardware described above withreference to step 406 of FIG. 4 (step 504). For example, the system mayestimate the fidelity of the quantum hardware by comparing the definedconvex combination with the determined output information of the quantumsystem and solving for the parameter α.

As described above, in some implementations the system may repeatedlyselect continuous Hamiltonian evolutions and repeatedly estimatecorresponding fidelities for the quantum hardware corresponding to thecontinuous Hamiltonian evolutions. By repeatedly estimating fidelitiesfor the quantum hardware, the system is able to increase the reliabilityof the fidelity estimate and reduce the likelihood of systematic errorsor correlations affecting obtained results. As a simple example, byrepeatedly estimating fidelities, the system is able to determine thatthe quantum hardware is functioning as intended.

Implementations of the digital and/or quantum subject matter and thedigital functional operations and quantum operations described in thisspecification can be implemented in digital electronic circuitry,suitable quantum circuitry or, more generally, quantum computationalsystems, in tangibly-embodied digital and/or quantum computer softwareor firmware, in digital and/or quantum computer hardware, including thestructures disclosed in this specification and their structuralequivalents, or in combinations of one or more of them. The term“quantum computational systems” may include, but is not limited to,quantum computers, quantum information processing systems, quantumcryptography systems, or quantum simulators.

Implementations of the digital and/or quantum subject matter describedin this specification can be implemented as one or more digital and/orquantum computer programs, e.g., one or more modules of digital and/orquantum computer program instructions encoded on a tangiblenon-transitory storage medium for execution by, or to control theoperation of, data processing apparatus. The digital and/or quantumcomputer storage medium can be a machine-readable storage device, amachine-readable storage substrate, a random or serial access memorydevice, one or more qubits, or a combination of one or more of them.Alternatively or in addition, the program instructions can be encoded onan artificially-generated propagated signal that is capable of encodingdigital and/or quantum information, e.g., a machine-generatedelectrical, optical, or electromagnetic signal, that is generated toencode digital and/or quantum information for transmission to suitablereceiver apparatus for execution by a data processing apparatus.

The terms quantum information and quantum data refer to information ordata that is carried by, held or stored in quantum systems, where thesmallest non-trivial system is a qubit, e.g., a system that defines theunit of quantum information. It is understood that the term “qubit”encompasses all quantum systems that may be suitably approximated as atwo-level system in the corresponding context. Such quantum systems mayinclude multi-level systems, e.g., with two or more levels. By way ofexample, such systems can include atoms, electrons, photons, ions orsuperconducting qubits. In many implementations the computational basisstates are identified with the ground and first excited states, howeverit is understood that other setups where the computational states areidentified with higher level excited states are possible. The term “dataprocessing apparatus” refers to digital and/or quantum data processinghardware and encompasses all kinds of apparatus, devices, and machinesfor processing digital and/or quantum data, including by way of examplea programmable digital processor, a programmable quantum processor, adigital computer, a quantum computer, multiple digital and quantumprocessors or computers, and combinations thereof. The apparatus canalso be, or further include, special purpose logic circuitry, e.g., anFPGA (field programmable gate array), an ASIC (application-specificintegrated circuit), or a quantum simulator, e.g., a quantum dataprocessing apparatus that is designed to simulate or produce informationabout a specific quantum system. In particular, a quantum simulator is aspecial purpose quantum computer that does not have the capability toperform universal quantum computation. The apparatus can optionallyinclude, in addition to hardware, code that creates an executionenvironment for digital and/or quantum computer programs, e.g., codethat constitutes processor firmware, a protocol stack, a databasemanagement system, an operating system, or a combination of one or moreof them.

A digital computer program, which may also be referred to or describedas a program, software, a software application, a module, a softwaremodule, a script, or code, can be written in any form of programminglanguage, including compiled or interpreted languages, or declarative orprocedural languages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, or other unitsuitable for use in a digital computing environment. A quantum computerprogram, which may also be referred to or described as a program,software, a software application, a module, a software module, a script,or code, can be written in any form of programming language, includingcompiled or interpreted languages, or declarative or procedurallanguages, and translated into a suitable quantum programming language,or can be written in a quantum programming language, e.g., QCL orQuipper.

A digital and/or quantum computer program may, but need not, correspondto a file in a file system. A program can be stored in a portion of afile that holds other programs or data, e.g., one or more scripts storedin a markup language document, in a single file dedicated to the programin question, or in multiple coordinated files, e.g., files that storeone or more modules, sub-programs, or portions of code. A digital and/orquantum computer program can be deployed to be executed on one digitalor one quantum computer or on multiple digital and/or quantum computersthat are located at one site or distributed across multiple sites andinterconnected by a digital and/or quantum data communication network. Aquantum data communication network is understood to be a network thatmay transmit quantum data using quantum systems, e.g. qubits. Generally,a digital data communication network cannot transmit quantum data,however a quantum data communication network may transmit both quantumdata and digital data.

The processes and logic flows described in this specification can beperformed by one or more programmable digital and/or quantum computers,operating with one or more digital and/or quantum processors, asappropriate, executing one or more digital and/or quantum computerprograms to perform functions by operating on input digital and quantumdata and generating output. The processes and logic flows can also beperformed by, and apparatus can also be implemented as, special purposelogic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or bya combination of special purpose logic circuitry or quantum simulatorsand one or more programmed digital and/or quantum computers.

For a system of one or more digital and/or quantum computers to be“configured to” perform particular operations or actions means that thesystem has installed on it software, firmware, hardware, or acombination of them that in operation cause the system to perform theoperations or actions. For one or more digital and/or quantum computerprograms to be configured to perform particular operations or actionsmeans that the one or more programs include instructions that, whenexecuted by digital and/or quantum data processing apparatus, cause theapparatus to perform the operations or actions. A quantum computer mayreceive instructions from a digital computer that, when executed by thequantum computing apparatus, cause the apparatus to perform theoperations or actions.

Digital and/or quantum computers suitable for the execution of a digitaland/or quantum computer program can be based on general or specialpurpose digital and/or quantum processors or both, or any other kind ofcentral digital and/or quantum processing unit. Generally, a centraldigital and/or quantum processing unit will receive instructions anddigital and/or quantum data from a read-only memory, a random accessmemory, or quantum systems suitable for transmitting quantum data, e.g.photons, or combinations thereof.

The essential elements of a digital and/or quantum computer are acentral processing unit for performing or executing instructions and oneor more memory devices for storing instructions and digital and/orquantum data. The central processing unit and the memory can besupplemented by, or incorporated in, special purpose logic circuitry orquantum simulators. Generally, a digital and/or quantum computer willalso include, or be operatively coupled to receive digital and/orquantum data from or transfer digital and/or quantum data to, or both,one or more mass storage devices for storing digital and/or quantumdata, e.g., magnetic, magneto-optical disks, optical disks, or quantumsystems suitable for storing quantum information. However, a digitaland/or quantum computer need not have such devices.

Digital and/or quantum computer-readable media suitable for storingdigital and/or quantum computer program instructions and digital and/orquantum data include all forms of non-volatile digital and/or quantummemory, media and memory devices, including by way of examplesemiconductor memory devices, e.g., EPROM, EEPROM, and flash memorydevices; magnetic disks, e.g., internal hard disks or removable disks;magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems,e.g., trapped atoms or electrons. It is understood that quantum memoriesare devices that can store quantum data for a long time with highfidelity and efficiency, e.g., light-matter interfaces where light isused for transmission and matter for storing and preserving the quantumfeatures of quantum data such as superposition or quantum coherence.

Control of the various systems described in this specification, orportions of them, can be implemented in a digital and/or quantumcomputer program product that includes instructions that are stored onone or more non-transitory machine-readable storage media, and that areexecutable on one or more digital and/or quantum processing devices. Thesystems described in this specification, or portions of them, can eachbe implemented as an apparatus, method, or system that may include oneor more digital and/or quantum processing devices and memory to storeexecutable instructions to perform the operations described in thisspecification.

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of what may beclaimed, but rather as descriptions of features that may be specific toparticular implementations. Certain features that are described in thisspecification in the context of separate implementations can also beimplemented in combination in a single implementation. Conversely,various features that are described in the context of a singleimplementation can also be implemented in multiple implementationsseparately or in any suitable sub-combination. Moreover, althoughfeatures may be described above as acting in certain combinations andeven initially claimed as such, one or more features from a claimedcombination can in some cases be excised from the combination, and theclaimed combination may be directed to a sub-combination or variation ofa sub-combination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various system modulesand components in the implementations described above should not beunderstood as requiring such separation in all implementations, and itshould be understood that the described program components and systemscan generally be integrated together in a single software product orpackaged into multiple software products.

Particular implementations of the subject matter have been described.Other implementations are within the scope of the following claims. Forexample, the actions recited in the claims can be performed in adifferent order and still achieve desirable results. As one example, theprocesses depicted in the accompanying figures do not necessarilyrequire the particular order shown, or sequential order, to achievedesirable results. In some cases, multitasking and parallel processingmay be advantageous.

The invention claimed is:
 1. A method comprising: accessing a set ofquantum gates; sampling a subset of quantum gates from the set ofquantum gates, wherein the subset of quantum gates defines a quantumcircuit; applying the quantum circuit to a quantum system and performingmeasurements on the quantum system to determine output information ofthe quantum system; calculating output information of the quantum systembased on application of the quantum circuit to the quantum system; andestimating a fidelity of the quantum circuit based on the determinedoutput information and the calculated output information of the quantumsystem, comprising fitting the determined output information of thequantum system to the calculated output information of the quantumsystem, wherein fitting the determined output information of the quantumsystem to the calculated output information of the quantum system toestimate the fidelity of the quantum circuit comprises: defining aconvex combination of the calculated output information of the quantumsystem and a totally mixed quantum state; and estimating the fidelity ofthe quantum circuit by comparing the defined convex combination with thedetermined output information of the quantum system.
 2. The method ofclaim 1, wherein the convex combination is given by ρ = α ⁢  ψ 〉 ⁢ 〈 ψ + ( 1 - α ) ⁢ N wherein α represents the fidelity of the quantumcircuit, |ψ

represents a calculated quantum state of the quantum system based onapplication of the quantum circuit to the quantum system, and

/N represents the totally mixed state.
 3. The method of claim 1, furthercomprising: repeatedly sampling a subset of quantum gates from the setof quantum gates until completion of an event, wherein each subset ofquantum gates defines a respective quantum circuit; for each sampledsubset of quantum gates: applying the respective quantum circuit to aquantum system and performing respective measurements on the quantumsystem to determine output information of the quantum system;calculating output information of the quantum system based onapplication of the respective quantum circuit to the quantum system; andestimating a fidelity of the respective quantum circuit based on thedetermined output information and the calculated output information ofthe quantum system.
 4. The method of claim 3, wherein the completion ofthe event occurs when an uncertainty of an estimated fidelity is below apredetermined threshold.
 5. The method of claim 1, wherein the set ofquantum gates comprises a universal set of quantum gates.
 6. The methodof claim 1, wherein the set of quantum gates comprise single qubitquantum gates and two qubit quantum gates.
 7. The method of claim 1,wherein each gate in the set of quantum gates is associated with arespective quantum gate fidelity.
 8. The method of claim 7, wherein thesampled subset of quantum gates comprises a same number of quantum gatesof comparable respective quantum gate fidelity.
 9. The method of claim1, wherein sampling a subset of quantum gates from the set of quantumgates comprises randomly sampling a subset of quantum gates from the setof quantum gates.
 10. An apparatus comprising: quantum hardwarecomprising: one or more qubits; one or more quantum gates; one or moremeasurement devices; one or more classical processors in datacommunication with the quantum hardware; wherein the quantum hardwareand the one or more classical processors are configured to performoperations comprising: accessing a set of quantum gates; sampling asubset of quantum gates from the set of quantum gates, wherein thesubset of quantum gates defines a quantum circuit; applying the quantumcircuit to a quantum system and performing measurements on the quantumsystem to determine output information of the quantum system;calculating output information of the quantum system based onapplication of the quantum circuit to the quantum system; and estimatinga fidelity of the quantum circuit based on the determined outputinformation and the calculated output information of the quantum system,comprising fitting the determined output information of the quantumsystem to the calculated output information of the quantum system,wherein fitting the determined output information of the quantum systemto the calculated output information of the quantum system to estimatethe fidelity of the quantum circuit comprises: defining a convexcombination of the calculated output information of the quantum systemand a totally mixed quantum state; and estimating the fidelity of thequantum circuit by comparing the defined convex combination with thedetermined output information of the quantum system.
 11. The apparatusof claim 10, wherein the one or more qubits are superconducting qubits.12. The apparatus of claim 10, wherein the one or more qubits form a onedimensional array.
 13. The apparatus of claim 10, wherein the one ormore qubits form a two dimensional array.
 14. The apparatus of claim 10,wherein each of the one or more qubits are subject to nearest neighborinteractions.